The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A292865 Determinants of the symmetric matrices whose entries on and below the main diagonal correspond to those of Pascal's triangle. 0
 1, 0, -1, 8, -71, 656, -4816, 1920, 168784, 43920880, -3315147449, 209095006856, -19095123359744, 1814464114046976, 320005209305667584, -253215321875947192320, -3298397219599339984896, 24417272707694829159671808, 265094852554176756050442657024, -931723550682987095264656018072440 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS FORMULA a(n) = det(L_n+U_n-I_{n+1}), where L_n is the lower triangular Pascal matrix of order n, U_n is the transpose of L_n and I_n is the identity matrix of order n. Note that L_n, U_n and I_n all have determinant 1 for all n. EXAMPLE a(0) is the determinant of the 1 X 1 matrix whose sole entry is one. a(1) is the determinant of the 2 X 2 matrix of all ones. a(2) is the determinant of the 3 X 3 matrix [1 1 1] [1 1 2] [1 2 1]. a(3) is the determinant of the 4 X 4 matrix [1 1 1 1] [1 1 2 3] [1 2 1 3] [1 3 3 1]. MATHEMATICA PascalMatrix = Function[n, Table[Table[Binomial[m, i], {i, 0, n}], {m, 0, n}]]; PascalDet = Function[n, Det[PascalMatrix[n] + Transpose[PascalMatrix[n]] - IdentityMatrix[n + 1]]]; Table[PascalDet[i], {i, 0, 19}] PROG (Python) from sympy import * def m(N):     return Matrix([         ([binomial(i, n) for n in range(i+1)] +[0] * (N-i))         for i in range(N+1)     ]) def matrix(N):     return m(N) + m(N).transpose() - eye(N+1) [ matrix(i).det() for i in range(20)] # Gilles Castel, Sep 25 2017 CROSSREFS Sequence in context: A015576 A070998 A187709 * A152265 A081178 A096341 Adjacent sequences:  A292862 A292863 A292864 * A292866 A292867 A292868 KEYWORD sign AUTHOR Alexander Farrugia, Sep 25 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified February 19 22:36 EST 2020. Contains 332061 sequences. (Running on oeis4.)